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Part I. Essays in Linear Algebra: 1. Perfect bases for the four subspaces; 2. Four subspaces from a graph; 3. My favourite matrix; 4. The ultimate in echelon forms; 5. Diagonalizing a symmetric matrix; 6. Pascal’s triangle in three matrices; 7. Can the factors be sparse?; 8. The eigenvalues of AB + BA; 9. Factoring into very special matrices; 10. Toeplitz matrices and circulants; 11. The joint spectral radius of A and B; 12. Banded matrices have remarkable inverses; Part II. Essays in Calculus: 13. The fundamental theorem of calculus; 14. The great function of calculus; 15. Thin strips and small triangles in calculus; 16. Chaos in Newton’s method; 17. Optical illusions for y = sin n; Part III. Essays in Applied Mathematics: 18. The principles of applied mathematics; 19. A minimum equals a maximum; 20. Aiming directly at the solution; 21. A chair through a door; Part IV. Essays in Computational Science: 22. Eight cosine transforms; 23. The ideas behind finite elements; 24. The Strang–Fix conditions; 25. Wavelets in music and television; 26. Scaling functions lead to wavelets; 27. Old and new splitting methods; An essay about books; List of publications; Index.

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